Selfdual Einstein hermitian four-manifolds

Vestislav Apostolov; Paul Gauduchon

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 1, page 203-243
  • ISSN: 0391-173X

Abstract

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We provide a local classification of selfdual Einstein riemannian four-manifolds admitting a positively oriented hermitian structure and characterize those which carry a hyperhermitian, non-hyperkähler structure compatible with the negative orientation. We show that selfdual Einstein 4-manifolds obtained as quaternionic quotients of P 2 and H 2 are hermitian.

How to cite

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Apostolov, Vestislav, and Gauduchon, Paul. "Selfdual Einstein hermitian four-manifolds." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.1 (2002): 203-243. <http://eudml.org/doc/84465>.

@article{Apostolov2002,
abstract = {We provide a local classification of selfdual Einstein riemannian four-manifolds admitting a positively oriented hermitian structure and characterize those which carry a hyperhermitian, non-hyperkähler structure compatible with the negative orientation. We show that selfdual Einstein 4-manifolds obtained as quaternionic quotients of $\{\mathbb \{H\}\}P^2$ and $\{\mathbb \{H\}\} H^2$ are hermitian.},
author = {Apostolov, Vestislav, Gauduchon, Paul},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {203-243},
publisher = {Scuola normale superiore},
title = {Selfdual Einstein hermitian four-manifolds},
url = {http://eudml.org/doc/84465},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Apostolov, Vestislav
AU - Gauduchon, Paul
TI - Selfdual Einstein hermitian four-manifolds
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 1
SP - 203
EP - 243
AB - We provide a local classification of selfdual Einstein riemannian four-manifolds admitting a positively oriented hermitian structure and characterize those which carry a hyperhermitian, non-hyperkähler structure compatible with the negative orientation. We show that selfdual Einstein 4-manifolds obtained as quaternionic quotients of ${\mathbb {H}}P^2$ and ${\mathbb {H}} H^2$ are hermitian.
LA - eng
UR - http://eudml.org/doc/84465
ER -

References

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  1. [1] V. Apostolov – J. Davidov – O. Muškarov, Compact Self-dual Hermitian Surfaces, Trans. Amer. Math. Soc. 348 (1996), 3051-3063. Zbl0880.53053MR1348147
  2. [2] V. Apostolov – P. Gauduchon, The Riemannian Goldberg-Sachs Theorem, Int. J. Math. 8 (1997), 421-439. Zbl0891.53054MR1460894
  3. [3] J. Armstrong, “Almost Kähler Geometry”, Ph.D. Thesis, Oxford, 1998. 
  4. [4] M. F. Atiyah – N. J. Hitchin – I. M. Singer, Self-duality in four dimensional geometry, Proc. Roy. Soc. London, A 362 (1979), 425-461. Zbl0389.53011MR506229
  5. [5] L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d’Einstein, Publications de l’Institut É. Cartan (Nancy) 4 (1982), 1-60. Zbl0544.53038
  6. [6] A. L. Besse, “Einstein manifolds”, Ergeb. Math. Grenzgeb.3, Folge 10, Springer-Verlag, Berlin, Heidelberg, New York, 1987. Zbl0613.53001MR867684
  7. [7] J.-P. Bourguignon, Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent. Math. 63 (1981), 263-286. Zbl0456.53033MR610539
  8. [8] C. Boyer – J. Finley, Killing vectors in self-dual Euclidean Einstein spaces, J. Math. Phys. (1982), 1126-1130. Zbl0484.53051MR660020
  9. [9] C. Boyer, Conformal duality and compact complex surfaces, Math. Ann. 274 (1986), 517-526. Zbl0571.32017MR842629
  10. [10] C. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), 157-164. Zbl0642.53073MR915736
  11. [11] C. Boyer, Self-dual and anti-self-dual Hermitian metrics on compact complex surfaces, In “Mathematics and General Relativity”, Proceedings, Santa Cruz 1986, J. Isenberg (ed.), Contemp. Math. 71 (1988), 105-114. Zbl0647.53053MR954411
  12. [12] R. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001) 623-715. Zbl1006.53019MR1824987
  13. [13] D. M. J. Calderbank, The Faraday 2-form in Einstein-Weyl geometry, Math. Scand. 89 (2001) 97-116. Zbl1130.53303MR1856983
  14. [14] D. M. J. Calderbank, The geometry of the Toda equation, J. Geom. Phys. 36 (2000) 152-162. Zbl0979.53046MR1783689
  15. [15] D. M. J. Calderbank, “Selfdual Einstein metrics and conformal submersions”, Edinburgh Preprint MS-00-001 (2000), available at arXiv: math.DG/0001041. 
  16. [16] D. M. J. Calderbank – K. P. Tod, Einstein metrics, hyper-complex structures and the Toda field equation, Differential Geom. Appl. 14 (2001) 199-208. Zbl1031.53071MR1822054
  17. [17] D. M. J. Calderbank – H. Pedersen, Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, Ann. Inst. Fourier 50 (2000), 921-963. Zbl0970.53027MR1779900
  18. [18] D. M. J. Calderbank – H. Pedersen, “Selfdual Einstein metrics with torus symmetry”, Edinburgh Preprint MS-00-022 (2000), available at arXiv:math.DG/0105263. Zbl1067.53034MR1950174
  19. [19] B.-Y. Chen, Some topological obstructions to Bochner-Kähler metrics and their applications, J. Differential Geom. 13 (1978), 574-588. Zbl0354.53049MR570217
  20. [20] A. Dancer – I. Strachan, Kähler-Einstein metrics with SU(2) action, Math. Proc. Cambridge Philos. Soc. 115 (1994), 513-525. Zbl0811.53046MR1269936
  21. [21] A. Derdziński, Exemples de métriques de Kähler et d’Einstein autoduales sur le plan complexe, In: “Géometrie riemannienne en dimension 4”, Séminaire Arthur Besse, L. Bérard-Bergery, M. Berger C. Houzel (eds.), CEDIC/Fernand Nathan, 1981. Zbl0477.53025
  22. [22] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405-433. Zbl0527.53030MR707181
  23. [23] M. G. Eastwood – K. P. Tod, Local constraints on Einstein-Weyl geometries, J. reine angew. Math. 491 (1997), 183-198. Zbl0876.53029MR1476092
  24. [24] K. Galicki, A generalization of the momentum mapping construction for quaternionic-Kähler manifolds, Comm. Math. Phys. 108 (1987), 108, 117-138. Zbl0608.53058MR872143
  25. [25] K. Galicki, New metrics with S p n S p 1 holonomy, Nucl. Phys. B 289 (1987), 573. 
  26. [26] K. Galicki – H. B. Lawson, Quaternionic Reduction and Quaternionic Orbifolds, Math. Ann. 282 (1988), 1-21. Zbl0628.53060MR960830
  27. [27] P. Gauduchon, La 1 -forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), 495-518. Zbl0523.53059MR742896
  28. [28] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1 × S 3 , J. reine angew. Math. 469 (1995), 1-50. Zbl0858.53039MR1363825
  29. [29] P. Gauduchon, Complex structures on compact conformal manifolds of negative type, In: “Complex Analysis and Geometry”, V. Ancona, E. Ballico and A. Silva (eds.), Marcel Dekker, New York-Basel-Hong Kong, 1996, 201-212. Zbl0870.53029MR1365975
  30. [30] P. Gauduchon, Connexion canonique et structures de Weyl en géométrie conforme, Preprint (unpublished). 
  31. [31] P. Gauduchon – K. P. Tod, Hyper-Hermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291-304. Zbl0945.53042MR1619847
  32. [32] G. Gibbons – S. Hawking, Classification of Gravitational Instanton Symmetries, Comm. Math. Phys. 66 (1979), 291-310. MR535152
  33. [33] M. Itoh, Self-duality of Kähler surfaces, Compositio Math. 51 (1984), 265-273. Zbl0546.53044MR739738
  34. [34] P. E. Jones – K. P. Tod, Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav. 2 (1985), 565-577. Zbl0575.53042MR795102
  35. [35] D. Joyce, Explicit construction of self-dual 4 -manifolds, Duke Math. J. 77 (1995), 519-552. Zbl0855.57028MR1324633
  36. [36] C. LeBrun, Counter-example to the generalized positive action conjecture, Comm. Math. Phys. 118 (1988), 591-596. Zbl0659.53050MR962489
  37. [37] C. LeBrun, private communication. 
  38. [38] A. B. Madsen, Einstein-Weyl structures in the conformal classes of LeBrun metrics, Class. Quantum Grav. 14 (1997), 2635-2645. Zbl0899.53040MR1477813
  39. [39] P. Nurowski, “Einstein equations and Cauchy-Riemann geometry”, Ph.D. Thesis, SISSA/ ISAS, Trieste, 1993. 
  40. [40] H. Pedersen, Einstein metrics, spinning top motions and monopoles, Math. Ann. 274 (1986), 35-59. Zbl0566.53058MR834105
  41. [41] H. Pedersen – A. Swann, Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. reine angew. Math. 441 (1993), 99-113. Zbl0776.53027MR1228613
  42. [42] J. Plebañski – M. Przanowski, Hermite-Einstein four-dimensional manifolds with symmetry, Class. Quantum Grav. 15 (1998), 1721-1735. Zbl0937.53034MR1628004
  43. [43] M. Przanowski – B. Broda, Locally Kähler gravitational instantons, Acta Phys. Pol. B 14 (1983), 637-661. MR732971
  44. [44] S. Salamon, Special structures on four-manifolds, Riv. Mat. Univ. Parma 17 (4) (1991), 109-123. Zbl0796.53031MR1219803
  45. [45] S. Salamon, “Riemannian geometry and holonomy groups”, Pitman Research Notes in Mathematics Series, 201, New York, 1989. Zbl0685.53001MR1004008
  46. [46] K. P. Tod, Cohomogeneity-one metrics with self-dual Weyl tensor, In “Twistor Theory”, S. Huggett (ed.), Marcel Dekker, New York (1995), 171-184. Zbl0827.53017MR1306964
  47. [47] K. P. Tod, Scalar-flat Kähler and hyper-Kähler metrics from Painlevé-III, Class. Quantum Grav. 12 (1995), 1535-1547. Zbl0828.53061MR1344288
  48. [48] K. P. Tod, The S U ( ) -Toda field equation and special four-dimensional metrics, In: “Geometry and Physics”, J. E. Andrsen, J. Dupont, H. Pedersen and A. Swann (eds.), Marcel Dekker, New York (1997), 307-312. Zbl0876.53026MR1423177
  49. [49] F. Tricerri – L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. Zbl0484.53014MR626479
  50. [50] I. Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc. 262 (1980), 533-542. Zbl0446.53048MR586733
  51. [51] I. Vaisman, Some curvature prperties of complex surfaces, Ann. Mat. Pura Appl. 32 (1982), 1-18. Zbl0512.53058MR696036
  52. [52] S. Webster, On the pseudo-conformal geometry of a Kähler manifolds, Math. Z. 157 (1977), 265-270. Zbl0354.53022MR477122

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